Talk:Hermitian manifold

"E bar" Undefined?
In the "Formal definition" section, the notation $$\overline{E}$$ isn't explicitly defined or referenced anywhere. Presumably, this denotes the conjugate bundle? Some clarification here would help, or if someone can confirm that this was the intended meaning, I can try to edit it in. Dzackgarza (talk) 02:16, 29 July 2020 (UTC)

In the introduction, shouldn't the almost complex structure preserve the metric, rather than the other way around? —Preceding unsigned comment added by 130.102.0.171 (talk) 01:11, 16 January 2009 (UTC)

Distinction between Hermitian Metric and Hermitian Structure
I think this article is a bit confusing at the moment. Should we not make a distinction between a Hermitian metric (given by the formula in the text) and a Hermitian structure (that is a smoothly varying choice of Hermitian form)? We could then show how each was related to the other. It's a simple enough point, but potentially confusing for newcomers.78.151.55.190 (talk) 00:02, 7 November 2013 (UTC)

Topological manifold?
Shouldn't the intro say smooth manifold? How exactly does one define an almost complex structure on a topological manifold? -- Fropuff 03:40, 10 October 2006 (UTC)
 * You're right and I changed it. VectorPosse 06:49, 10 October 2006 (UTC)\
 * (Needless to say, every complex manifold is a topological manifold.) 2601:200:C082:2EA0:685C:5576:974E:96FD (talk) 20:15, 18 May 2023 (UTC)

Either confusing notation or completely wrong.
Either the notation in section 2 is very misleading, or many of the statements are completely wrong. A Hermitian $$n$$-by-$$n$$ matrix $$h$$ can indeed be decomposed into two parts $$ h=a+ib$$ where $$a=(h+\bar h)/2 $$ is a real, symetric  $$n$$-by-$$n$$ matrix and $$b=(h-\bar h)/2i $$ is a real skew-symmetric  $$n$$-by-$$n$$ matrix. The article imples that the metric $$g$$ is to be identified with $$a$$, and $$\omega$$ with $$ b$$. It then states that one can be obtained from the other by means of the complex structure $$J$$. This cannot be true. Hermiticity requires no relation between  $$ a$$ and $$b$$.

What is actually true is that there is a Riemann metric $$g$$ on the underlying $$2n$$-dimensional real smooth manifold, where

g=\left(\begin{matrix} a & b \\-b & a\end{matrix}\right) $$ is a real symmetric $$2n$$-by-$$2n$$ matrix, while

\omega =\left(\begin{matrix} b & -a \\a & b\end{matrix}\right)=\left(\begin{matrix} 0 & -I_n \\I_n & 0\end{matrix}\right) \left(\begin{matrix} a & b \\-b & a \end{matrix}\right)=\left(\begin{matrix} a & b \\-b & a\end{matrix}\right)\left(\begin{matrix} 0 & -I_n \\I_n & 0\end{matrix}\right) $$ is a skew-symmetric $$2n$$-by-$$2n$$ matrix. Here

J=\left(\begin{matrix} 0 & -I_n \\I_n & 0\end{matrix}\right) $$ is the $$2n$$-by-$$2n$$ matrix representing the complex structure in the underlying real vector space. $$J$$ commutes with $$g$$ because it is simply multiplication by $$\sqrt{-1}$$ in the original complex basis. Mike Stone (talk) 19:01, 9 December 2016 (UTC)

The $$\Gamma$$ in the "Formal definition" section is not defined
I think defining it explicitly would ease the understanding of the article. Luca (talk) 14:32, 11 February 2021 (UTC)

What does a bar over an index mean?
In multiple places in the article, a metric h is shown with subscripts 𝛼 and 𝛽, and the 𝛽 has a bar over it.

If the article explains this notation somewhere, I have not found where it might be.

Taking the complex conjugate of an *index* does not seem to be the explanation (for they are all integers, anyway). So: What does this mean?

I hope someone can include an explanation of this notation in the article ... or else rid the article of this unexplained notation by rewriting some sections.